{
  "type": "author mathematical validation",
  "exact_symbolic_identities": {
    "EX10_interval_width": true,
    "EX10_small_window_implication": true,
    "EX12_interval_width": true,
    "EX12_window_implication": true,
    "EX10_exact_circle_quadric": true,
    "universal_derivative_phase": true
  },
  "rational_cases": [
    {
      "family": 10,
      "a": 2,
      "b": 4,
      "c": "15/16",
      "lower": "7/12",
      "upper": "3/4",
      "gamma": "2/3",
      "k": "5/12",
      "ratio_exponent": "-1/4",
      "K_over_T_exponent": "-1/4",
      "height_over_TK_exponent": "-1/12",
      "eta_exponent": "-1/3",
      "small_window_exponent": "-5/24"
    },
    {
      "family": 10,
      "a": 4,
      "b": 7,
      "c": "27/32",
      "lower": "11/20",
      "upper": "7/12",
      "gamma": "17/30",
      "k": "29/60",
      "ratio_exponent": "-1/4",
      "K_over_T_exponent": "-1/12",
      "height_over_TK_exponent": "-1/20",
      "eta_exponent": "-13/30",
      "small_window_exponent": "-29/240"
    },
    {
      "family": 10,
      "a": 5,
      "b": 9,
      "c": "17/20",
      "lower": "7/12",
      "upper": "5/8",
      "gamma": "29/48",
      "k": "23/48",
      "ratio_exponent": "-1/2",
      "K_over_T_exponent": "-1/8",
      "height_over_TK_exponent": "-1/12",
      "eta_exponent": "-19/48",
      "small_window_exponent": "-23/240"
    },
    {
      "family": 12,
      "a": 4,
      "b": 4,
      "c": null,
      "lower": "3/5",
      "upper": "2/3",
      "gamma": "19/30",
      "k": "7/15",
      "ratio_exponent": "-1",
      "K_over_T_exponent": "-1/6",
      "height_over_TK_exponent": "-1/10",
      "eta_exponent": "-11/30",
      "small_window_exponent": "-7/15"
    },
    {
      "family": 12,
      "a": 6,
      "b": 5,
      "c": null,
      "lower": "4/7",
      "upper": "3/5",
      "gamma": "41/70",
      "k": "17/35",
      "ratio_exponent": "-1",
      "K_over_T_exponent": "-1/10",
      "height_over_TK_exponent": "-1/14",
      "eta_exponent": "-29/70",
      "small_window_exponent": "-17/35"
    },
    {
      "family": 12,
      "a": 7,
      "b": 6,
      "c": null,
      "lower": "5/8",
      "upper": "2/3",
      "gamma": "31/48",
      "k": "23/48",
      "ratio_exponent": "-1",
      "K_over_T_exponent": "-1/6",
      "height_over_TK_exponent": "-1/8",
      "eta_exponent": "-17/48",
      "small_window_exponent": "-23/48"
    }
  ],
  "bounded_integer_scan": {
    "a": [
      2,
      40
    ],
    "admissible_family10_pairs": 400,
    "admissible_family12_pairs": 361
  },
  "first_lower_order_pairs": {
    "family10": [
      4,
      7
    ],
    "family12": [
      6,
      5
    ]
  },
  "explicit_polynomial_signs": [
    {
      "family": 10,
      "a": 2,
      "limiting_coefficient": "8",
      "derivative_at_z0": "-16*I"
    },
    {
      "family": 10,
      "a": 4,
      "limiting_coefficient": "-32",
      "derivative_at_z0": "-128*I"
    },
    {
      "family": 10,
      "a": 5,
      "limiting_coefficient": "64*I",
      "derivative_at_z0": "-320*I"
    },
    {
      "family": 12,
      "a": 4,
      "limiting_coefficient": "-1",
      "derivative_at_z0": "-4*I"
    },
    {
      "family": 12,
      "a": 6,
      "limiting_coefficient": "1",
      "derivative_at_z0": "-6*I"
    },
    {
      "family": 12,
      "a": 7,
      "limiting_coefficient": "-I",
      "derivative_at_z0": "-7*I"
    }
  ],
  "scope": "Exact calculations validate displayed identities and chosen paths; the general proofs and relative theorem dependency scope remain in ../draft.md.",
  "sha256": {
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    "threshold-regions.svg": "B98B5E363B17987C476846F2ED1A2A951EEC7CCF61FFABA6FA6A84F3EFC3D439",
    "scale-intervals.png": "CD6160F95CEA3EEF85783903DC9C9DF6762982BE67FC03E4186B254669830496",
    "scale-intervals.svg": "A29606903E3E4AD0470B85C0409C0B4E9BC6D5146414DC0426D5A86694B637D0"
  }
}